Relevant bibliographies by topics / Tensors methods

Academic literature on the topic 'Tensors methods'

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Published: 7 July 2024

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Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Journal articles on the topic "Tensors methods":

1

Katoch, Nitish, Bup-Kyung Choi, Ji-Ae Park, In-Ok Ko, and Hyung-Joong Kim. "Comparison of Five Conductivity Tensor Models and Image Reconstruction Methods Using MRI." Molecules 26, no. 18 (September 10, 2021): 5499. http://dx.doi.org/10.3390/molecules26185499.

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Imaging of the electrical conductivity distribution inside the human body has been investigated for numerous clinical applications. The conductivity tensors of biological tissue have been obtained from water diffusion tensors by applying several models, which may not cover the entire phenomenon. Recently, a new conductivity tensor imaging (CTI) method was developed through a combination of B1 mapping, and multi-b diffusion weighted imaging. In this study, we compared the most recent CTI method with the four existing models of conductivity tensors reconstruction. Two conductivity phantoms were designed to evaluate the accuracy of the models. Applied to five human brains, the conductivity tensors using the four existing models and CTI were imaged and compared with the values from the literature. The conductivity image of the phantoms by the CTI method showed relative errors between 1.10% and 5.26%. The images by the four models using DTI could not measure the effects of different ion concentrations subsequently due to prior information of the mean conductivity values. The conductivity tensor images obtained from five human brains through the CTI method were comparable to previously reported literature values. The images by the four methods using DTI were highly correlated with the diffusion tensor images, showing a coefficient of determination (R2) value of 0.65 to 1.00. However, the images by the CTI method were less correlated with the diffusion tensor images and exhibited an averaged R2 value of 0.51. The CTI method could handle the effects of different ion concentrations as well as mobilities and extracellular volume fractions by collecting and processing additional B1 map data. It is necessary to select an application-specific model taking into account the pros and cons of each model. Future studies are essential to confirm the usefulness of these conductivity tensor imaging methods in clinical applications, such as tumor characterization, EEG source imaging, and treatment planning for electrical stimulation.
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Wang, Hai Jun, Fei Yun Xu, and Fei Wang. "Tensor Factorization and Clustering for the Feature Extraction Based on Tucker3 with Updating Core." Advanced Materials Research 308-310 (August 2011): 2517–22. http://dx.doi.org/10.4028/www.scientific.net/amr.308-310.2517.

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Aiming at the problems of Tucker3 to large-scale tensor when applied to feature extraction, a new factorization based on Tucker3 is proposed to extract feature from the tensors. First, the large-scale tensor is divided into multiple sub-tensors so as to conveniently compute cores of sub-tensors in parallel mode with Matlab Parallel Computing Toolbox; Then, the cores of each sub-tensor are updated for reducing deviation in calculating and the similar characteristics of sub-tensors are clustered to obtain the features. Experiment results show that this methods is able to extract features rapidly and efficiently.
3

Moes, H., E. G. Sikkes, and R. Bosma. "Mobility and Impedance Tensor Methods for Full and Partial-Arc Journal Bearings." Journal of Tribology 108, no. 4 (October 1, 1986): 612–19. http://dx.doi.org/10.1115/1.3261282.

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Mobility and impedance tensors are introduced for full journal bearings. These tensors may replace the well known mobility and impedance vectors. Since tensors apply to arbitrary systems of reference, coordinates rotating with the sleeve, i.e. fixed coordinates, will be introduced and will henceforth replace the unidirectional system in current use. As a consequence, the restriction to full journal bearing applications which was necessary up to now may be withdrawn. The mobility- and impedance methods, as derived for full journal bearings, from now on apply equally well to partial-arc bearings. The mobility- and impedance tensor descriptions, needed in applications of the methods, can be derived in a straightforward manner form the generally applied vector descriptions for full journal bearings. Descriptions for partial-arc bearings will also be presented.
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Yang, Hye-Kyung, and Hwan-Seung Yong. "Multi-Aspect Incremental Tensor Decomposition Based on Distributed In-Memory Big Data Systems." Journal of Data and Information Science 5, no. 2 (May 20, 2020): 13–32. http://dx.doi.org/10.2478/jdis-2020-0010.

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AbstractPurposeWe propose InParTen2, a multi-aspect parallel factor analysis three-dimensional tensor decomposition algorithm based on the Apache Spark framework. The proposed method reduces re-decomposition cost and can handle large tensors.Design/methodology/approachConsidering that tensor addition increases the size of a given tensor along all axes, the proposed method decomposes incoming tensors using existing decomposition results without generating sub-tensors. Additionally, InParTen2 avoids the calculation of Khari–Rao products and minimizes shuffling by using the Apache Spark platform.FindingsThe performance of InParTen2 is evaluated by comparing its execution time and accuracy with those of existing distributed tensor decomposition methods on various datasets. The results confirm that InParTen2 can process large tensors and reduce the re-calculation cost of tensor decomposition. Consequently, the proposed method is faster than existing tensor decomposition algorithms and can significantly reduce re-decomposition cost.Research limitationsThere are several Hadoop-based distributed tensor decomposition algorithms as well as MATLAB-based decomposition methods. However, the former require longer iteration time, and therefore their execution time cannot be compared with that of Spark-based algorithms, whereas the latter run on a single machine, thus limiting their ability to handle large data.Practical implicationsThe proposed algorithm can reduce re-decomposition cost when tensors are added to a given tensor by decomposing them based on existing decomposition results without re-decomposing the entire tensor.Originality/valueThe proposed method can handle large tensors and is fast within the limited-memory framework of Apache Spark. Moreover, InParTen2 can handle static as well as incremental tensor decomposition.
5

Moore, J. G., S. A. Schorn, and J. Moore. "Education Committee Best Paper of 1995 Award: Methods of Classical Mechanics Applied to Turbulence Stresses in a Tip Leakage Vortex." Journal of Turbomachinery 118, no. 4 (October 1, 1996): 622–29. http://dx.doi.org/10.1115/1.2840917.

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Moore et al. measured the six Reynolds stresses in a tip leakage vortex in a linear turbine cascade. Stress tensor analysis, as used in classical mechanics, has been applied to the measured turbulence stress tensors. Principal directions and principal normal stresses are found. A solid surface model, or three-dimensional glyph, for the Reynolds stress tensor is proposed and used to view the stresses throughout the tip leakage vortex. Modeled Reynolds stresses using the Boussinesq approximation are obtained from the measured mean velocity strain rate tensor. The comparison of the principal directions and the three-dimensional graphic representations of the strain and Reynolds stress tensors aids in the understanding of the turbulence and what is required to model it.
6

Xue, Zhaohui, Sirui Yang, Hongyan Zhang, and Peijun Du. "Coupled Higher-Order Tensor Factorization for Hyperspectral and LiDAR Data Fusion and Classification." Remote Sensing 11, no. 17 (August 21, 2019): 1959. http://dx.doi.org/10.3390/rs11171959.

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Hyperspectral and light detection and ranging (LiDAR) data fusion and classification has been an active research topic, and intensive studies have been made based on mathematical morphology. However, matrix-based concatenation of morphological features may not be so distinctive, compact, and optimal for classification. In this work, we propose a novel Coupled Higher-Order Tensor Factorization (CHOTF) model for hyperspectral and LiDAR data classification. The innovative contributions of our work are that we model different features as multiple third-order tensors, and we formulate a CHOTF model to jointly factorize those tensors. Firstly, third-order tensors are built based on spectral-spatial features extracted via attribute profiles (APs). Secondly, the CHOTF model is defined to jointly factorize the multiple higher-order tensors. Then, the latent features are generated by mode-n tensor-matrix product based on the shared and unshared factors. Lastly, classification is conducted by using sparse multinomial logistic regression (SMLR). Experimental results, conducted with two popular hyperspectral and LiDAR data sets collected over the University of Houston and the city of Trento, respectively, indicate that the proposed framework outperforms the other methods, i.e., different dimensionality-reduction-based methods, independent third-order tensor factorization based methods, and some recently proposed hyperspectral and LiDAR data fusion and classification methods.
7

Hajarian, Masoud. "Solving coupled tensor equations via higher order LSQR methods." Filomat 34, no. 13 (2020): 4419–27. http://dx.doi.org/10.2298/fil2013419h.

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Tensors have a wide application in control theory, data mining, chemistry, information sciences, documents analysis and medical engineering. The material here is motivated by the development of the efficient numerical methods for solving the coupled tensor equations (A1*M X *N B1 + C1 *M Y *N D1 = E1, A2 *M X *N B2 + C2 *M Y *N D2 = E2, with Einstein product. We propose the tensor form of the LSQR methods to find the solutions X and Y of the coupled tensor equations. Finally we give some numerical examples to illustrate that our proposed methods are able to accurately and efficiently find the solutions of tensor equations with Einstein product.
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Zhong, Guoqiang, and Mohamed Cheriet. "Large Margin Low Rank Tensor Analysis." Neural Computation 26, no. 4 (April 2014): 761–80. http://dx.doi.org/10.1162/neco_a_00570.

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We present a supervised model for tensor dimensionality reduction, which is called large margin low rank tensor analysis (LMLRTA). In contrast to traditional vector representation-based dimensionality reduction methods, LMLRTA can take any order of tensors as input. And unlike previous tensor dimensionality reduction methods, which can learn only the low-dimensional embeddings with a priori specified dimensionality, LMLRTA can automatically and jointly learn the dimensionality and the low-dimensional representations from data. Moreover, LMLRTA delivers low rank projection matrices, while it encourages data of the same class to be close and of different classes to be separated by a large margin of distance in the low-dimensional tensor space. LMLRTA can be optimized using an iterative fixed-point continuation algorithm, which is guaranteed to converge to a local optimal solution of the optimization problem. We evaluate LMLRTA on an object recognition application, where the data are represented as 2D tensors, and a face recognition application, where the data are represented as 3D tensors. Experimental results show the superiority of LMLRTA over state-of-the-art approaches.
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Shi, Qiquan, Jiaming Yin, Jiajun Cai, Andrzej Cichocki, Tatsuya Yokota, Lei Chen, Mingxuan Yuan, and Jia Zeng. "Block Hankel Tensor ARIMA for Multiple Short Time Series Forecasting." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 5758–66. http://dx.doi.org/10.1609/aaai.v34i04.6032.

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This work proposes a novel approach for multiple time series forecasting. At first, multi-way delay embedding transform (MDT) is employed to represent time series as low-rank block Hankel tensors (BHT). Then, the higher-order tensors are projected to compressed core tensors by applying Tucker decomposition. At the same time, the generalized tensor Autoregressive Integrated Moving Average (ARIMA) is explicitly used on consecutive core tensors to predict future samples. In this manner, the proposed approach tactically incorporates the unique advantages of MDT tensorization (to exploit mutual correlations) and tensor ARIMA coupled with low-rank Tucker decomposition into a unified framework. This framework exploits the low-rank structure of block Hankel tensors in the embedded space and captures the intrinsic correlations among multiple TS, which thus can improve the forecasting results, especially for multiple short time series. Experiments conducted on three public datasets and two industrial datasets verify that the proposed BHT-ARIMA effectively improves forecasting accuracy and reduces computational cost compared with the state-of-the-art methods.
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DE AZCÁRRAGA, J. A., and A. J. MACFARLANE. "COMPILATION OF RELATIONS FOR THE ANTISYMMETRIC TENSORS DEFINED BY THE LIE ALGEBRA COCYCLES OF su(n)." International Journal of Modern Physics A 16, no. 08 (March 30, 2001): 1377–405. http://dx.doi.org/10.1142/s0217751x01003111.

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This paper attempts to provide a comprehensive compilation of results, many new here, involving the invariant totally antisymmetric tensors (Omega tensors) which define the Lie algebra cohomology cocycles of su (n), and that play an essential role in the optimal definition of Racah–Casimir operators of su (n). Since the Omega tensors occur naturally within the algebra of totally antisymmetrized products of λ-matrices of su (n), relations within this algebra are studied in detail, and then employed to provide a powerful means of deriving important Omega tensor/cocycle identities. The results include formulas for the squares of all the Omega tensors of su (n). Various key derivations are given to illustrate the methods employed.
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Journal articles Dissertations / Theses Books

Dissertations / Theses on the topic "Tensors methods":

1

Handschuh, Stefan. "Numerical methods in Tensor Networks." Doctoral thesis, Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-159672.

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In many applications that deal with high dimensional data, it is important to not store the high dimensional object itself, but its representation in a data sparse way. This aims to reduce the storage and computational complexity. There is a general scheme for representing tensors with the help of sums of elementary tensors, where the summation structure is defined by a graph/network. This scheme allows to generalize commonly used approaches in representing a large amount of numerical data (that can be interpreted as a high dimensional object) using sums of elementary tensors. The classification does not only distinguish between elementary tensors and non-elementary tensors, but also describes the number of terms that is needed to represent an object of the tensor space. This classification is referred to as tensor network (format). This work uses the tensor network based approach and describes non-linear block Gauss-Seidel methods (ALS and DMRG) in the context of the general tensor network framework. Another contribution of the thesis is the general conversion of different tensor formats. We are able to efficiently change the underlying graph topology of a given tensor representation while using the similarities (if present) of both the original and the desired structure. This is an important feature in case only minor structural changes are required. In all approximation cases involving iterative methods, it is crucial to find and use a proper initial guess. For linear iteration schemes, a good initial guess helps to decrease the number of iteration steps that are needed to reach a certain accuracy, but it does not change the approximation result. For non-linear iteration schemes, the approximation result may depend on the initial guess. This work introduces a method to successively create an initial guess that improves some approximation results. This algorithm is based on successive rank 1 increments for the r-term format. There are still open questions about how to find the optimal tensor format for a given general problem (e.g. storage, operations, etc.). For instance in the case where a physical background is given, it might be efficient to use this knowledge to create a good network structure. There is however, no guarantee that a better (with respect to the problem) representation structure does not exist.
2

Wu, Yanqi. "New methods for measuring CSA tensors : applications to nucleotides and nucleosides." Thesis, University of Nottingham, 2011. http://eprints.nottingham.ac.uk/11859/.

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A novel version of the CSA (Chemical Shift Anisotropy) amplification experiment which results in large amplification factors is introduced. Large xa (up to 48) are achieved by sequences which are efficient in terms of the number of π pulses and total duration compared to a modification due to Orr et al. (2005), and greater flexibility in terms of the choice of amplification factor is possible than in our most recent version. Furthermore, the incorporation of XiX decoupling ensures the overall sensitivity of the experiment is optimal. This advantage has been proved by extracting the CSA tensors for a novel vinylphosphonate-linked nucleotide. The application of CSA amplification experiment to six nucleosides is also discussed. The measured principal tensor values are compared with those calculated using the recently developed first-principles methods. Throughout this work, the NMR parameters of all nucleosides are presented. Finally, high-resolution multi-nuclear solid-state NMR experiments are used to study some novel vinyl phosphonate-linked oligo-nucleotides.
3

Damodaran, K. "Spatially dependent interaction tensors determined through novel methods of high resolution solid state NMR." Thesis(Ph.D.), CSIR-National Chemical Laboratory, Pune, 2006. http://dspace.ncl.res.in:8080/xmlui/handle/20.500.12252/2493.

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Lund, Kathryn. "A new block Krylov subspace framework with applications to functions of matrices acting on multiple vectors." Diss., Temple University Libraries, 2018. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/493337.

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Mathematics
Ph.D.
We propose a new framework for understanding block Krylov subspace methods, which hinges on a matrix-valued inner product. We can recast the ``classical" block Krylov methods, such as O'Leary's block conjugate gradients, global methods, and loop-interchange methods, within this framework. Leveraging the generality of the framework, we develop an efficient restart procedure and error bounds for the shifted block full orthogonalization method (Sh-BFOM(m)). Regarding BFOM as the prototypical block Krylov subspace method, we propose another formalism, which we call modified BFOM, and show that block GMRES and the new block Radau-Lanczos method can be regarded as modified BFOM. In analogy to Sh-BFOM(m), we develop an efficient restart procedure for shifted BGMRES with restarts (Sh-BGMRES(m)), as well as error bounds. Using this framework and shifted block Krylov methods with restarts as a foundation, we formulate block Krylov subspace methods with restarts for matrix functions acting on multiple vectors f(A)B. We obtain convergence bounds for \bfomfom (BFOM for Functions Of Matrices) and block harmonic methods (i.e., BGMRES-like methods) for matrix functions. With various numerical examples, we illustrate our theoretical results on Sh-BFOM and Sh-BGMRES. We also analyze the matrix polynomials associated to the residuals of these methods. Through a variety of real-life applications, we demonstrate the robustness and versatility of B(FOM)^2 and block harmonic methods for matrix functions. A particularly interesting example is the tensor t-function, our proposed definition for the function of a tensor in the tensor t-product formalism. Despite the lack of convergence theory, we also show that the block Radau-Lanczos modification can reduce the number of cycles required to converge for both linear systems and matrix functions.
Temple University--Theses
5

Savas, Berkant. "Algorithms in data mining using matrix and tensor methods." Doctoral thesis, Linköpings universitet, Beräkningsvetenskap, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11597.

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In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors. The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is \[ \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp, \] where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix. Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations. The remaining two papers discuss computational methods for the best multilinear rank approximation problem \[ \min_{\cB} \| \cA - \cB\| \] where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product \[ \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu}, \] where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.
6

Flores, Philippe. "Estimation of high dimensional probability density functions with low rank-tensors models : application to flow cytometry." Electronic Thesis or Diss., Université de Lorraine, 2024. http://www.theses.fr/2024LORR0021.

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La cytométrie en flux (CMF) est une technique d'analyse de cellules biologiques largement utilisée en immunologie, par exemple dans la recherche sur les leucémies. Le principe de la CMF est de mesurer les propriétés de fluorescence individuellement dans un volume de cellules. L'analyse des données de CMF permet d'identifier et de caractériser les populations de cellules à l'intérieur d'un volume de cellules. Les analyses effectuées manuellement reposent sur la sélection de cellules sur des nuages de points bivariés. Cette opération, appelée gating, prend du temps et est subjective. Bien qu'il existe des méthodes non supervisées, ces méthodes prennent souvent beaucoup de temps et ne permettent pas de traiter de grands jeux de données. Pour analyser les jeux de données de CMF, nous avons décidé d'utiliser une approche probabiliste. Dans ce cas, le problème d'analyse de données de CMF revient à une estimation de densités de probabilité. Dans un chapitre préliminaire, nous présentons le problème d'estimation d'histogrammes multivariés. Ce problème est considéré comme impossible en pratique en raison de la malédiction de la dimension (MdD) affirmant que la complexité d'un problème augmente de façon exponentielle avec le nombre de dimensions. Pour résoudre ce problème, deux solutions sont proposées dans la littérature. Premièrement, la densité est modélisée avec un modèle Bayésien naïf (MBN) dont la complexité est linéaire avec le nombre de dimensions. Deuxièmement, les facteurs du MBN sont obtenus via un algorithme de factorisation tensorielle couplée. Cette méthode, appelée CTF3D, couple des marginales 3D qui sont faciles à calculer en CMF par exemple. Cependant, CTF3D n'a pas résolu la MdD mais l'a plutôt déplacée à un autre niveau : le nombre de marginales 3D. Nous proposons alors un algorithme résolvant le troisième niveau de MdD. Cette méthode appelée PCTF3D couple des sous-ensembles de marginales 3D. En choisissant un sous-ensemble de triplets et donc le nombre de triplets, la complexité de PCTF3D est réduite et contrôlée par l'utilisateur. Le choix des triplets est appelé une stratégie de couplage et différentes stratégies sont présentées sous la forme d'hypergraphes. Par exemple, les stratégies aléatoires consistent à choisir des triplets au hasard alors que les stratégies équilibrées consistent à choisir des triplets de telle sorte que toutes les variables soient représentées de manière égale. Un algorithme de génération de couplages équilibrés est proposé. Enfin, des expériences numériques sur des ensembles de données réelles et synthétiques sont réalisées. Notre nouvelle méthode a introduit un modèle couplé de tenseur. Dans le quatrième chapitre, nous abordons le problème d'unicité de ce nouveau modèle. Tout d'abord, la recouvrabilité est étudiée et un algorithme qui trouve la borne de recouvrabilité est présenté. Il est basé sur l'étude du rang de la jacobienne de la paramétrisation. Quand il est appliqué à des couplages aléatoires, des cas défectueux sont observés et conduisent à des baisses de bornes de recouvrabilité. Ces cas ne sont pas observés pour les couplages équilibrés, ce qui en fait une bonne alternative pour garantir l'unicité du modèle. Deuxièmement, l'identifiabilité du modèle a été examinée. Nous utilisons les preuves précédemment démontrées dans la littérature pour prouver de meilleures conditions suffisantes d'identifiabilité. Enfin, notre méthode d'estimation est utilisée pour l'analyse de données de CMF. En considérant un MBN pour la distribution des cellules, les facteurs du MBN sont interprétés comme des groupes de cellules représentés par une proportion et leurs propriétés de fluorescence. Cette méthode nommée CTFlowHD utilise PCTF3D pour obtenir les facteurs du MBN. Après cette étape, nous présentons plusieurs outils pour visualiser les termes de rang 1. Notre méthode permet d'utiliser divers outils de visualisation, en particulier des outils déjà utilisés dans la communauté FCM
Flow cytometry (FCM) is one of a most popular techniques for biological cells analysis. It is widely used in immunology, where it permits to make advances in leukemia research for example. The principle of FCM is to measure fluorescence properties for each cell present in a volume of cells. FCM data analysis permit to identify and characterize cell populations. Analysis performed manually rely on selection of cells plotted with bivariate plots. This operation, called gating, is time-consuming and introduces subjectivity. Although unsupervised methods exists, they are time-consuming and does not handle large datasets. To analyze FCM datasets, we decided to use a probabilistic approach. In that sense, the problem of FCM data analysis comes to an estimation of a probability density function. In the second and preliminary chapter, we present the problem of multivariate histogram estimation. This problem is considered impossible in practice because of the Curse of Dimensionality (CoD) which states that the complexity of a problem increases exponentially with the number of dimensions. To solve this issue, two solutions are performed in the litterature. First, the density is modelled with a naive Bayes model (NBM) whose complexity remains linear with the number of dimensions. Secondly, the factors of the NBM are obtained via a coupled tensor factorization algorithm. This method called CTF3D couples 3D marginals which are easy to compute with the amount of data available in FCM. However, CTF3D did not fully solved the CoD but instead moved it to another level: the number of 3D marginals. In a third chapter, we propose a new algorithm that solves the third level of CoD. This method called Partial Coupled Tensor Factorization of 3D marginals or PCTF3D is coupling subsets of 3D marginals. By choosing a subset of triplets hence the number of triplets, PCTF3D's complexity is reduced and controlled by end-users. The choice of triplets is called a coupling strategy and different strategies are presented with the formalism of hypergraphs. For example, random strategies consists in choosing triplets randomly. Balanced strategies consists in choosing triplets such that variables are represented evenly. An algorithm for balanced coupling generation is proposed. Finally, numerical experiments on real and synthetic datasets are performed. Our new method introduced a partially coupled tensor model. In the fourth chapter, we address the problem of uniqueness of this new model. First, recoverability is studied and an algorithm that finds the recoverability bound is presented. This algorithm is based on the study of the rank of the Jacobian of the parametrization. When applied to random couplings, defective cases are observed which leads to drops in recoverability bounds. Those cases are not observed for balanced couplings, making this strategy a good alternative for uniqueness guarantees. In a second part, the identifiability of the model was examined. We use previous proofs to demonstrate new identifiability sufficient conditions that exceed the conditions of the litterature. At last, our new histogram estimation method was used for FCM data analysis. We present Coupled Tensor factorization for Flow cytometry in High Dimensions or CTFlowHD: an unsupervised workflow for FCM data analysis. By considering a NBM for the cell distribution, NBM component are interpreted as cell groups which are represented by a proportion and a set of fluorescence properties. CTFlowHD uses PCTF3D to obtain the factors of the NBM. After this step, we present several tools for visualizing the rank-one terms. The main advantage of our method is that the visualization step can be applied without having to compute the NBM factors again. This permits to use various tools for visualization, especially tools already used in the FCM community. Finally, CTFlowHD is applied to real datasets
7

Bridgeman, Jacob. "Tensor Network Methods for Quantum Phases." Thesis, The University of Sydney, 2017. http://hdl.handle.net/2123/17647.

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The physics that emerges when large numbers of particles interact can be complex and exotic. The collective behaviour may not re ect the underlying constituents, for example fermionic quasiparticles can emerge from models of interacting bosons. Due to this emergent complexity, manybody phenomena can be very challenging to study, but also very useful. A theoretical understanding of such systems is important for robust quantum information storage and processing. The emergent, macroscopic physics can be classi ed using the idea of a quantum phase. All models within a given phase exhibit similar low-energy emergent physics, which is distinct from that displayed by models in di erent phases. In this thesis, we utilise tensor networks to study many-body systems in a range of quantum phases. These include topologically ordered phases, gapless symmetry-protected phases, and symmetry-enriched topological phases.
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Gomes, Paulo Ricardo Barboza. "Tensor Methods for Blind Spatial Signature Estimation." Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=11635.

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FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico
In this dissertation the problem of spatial signature and direction of arrival estimation in Linear 2L-Shape and Planar arrays is investigated Methods based on tensor decompositions are proposed to treat the problem of estimating blind spatial signatures disregarding the use of training sequences and knowledge of the covariance structure of the sources By assuming that the power of the sources varies between successive time blocks decompositions for tensors of third and fourth orders obtained from spatial and spatio-temporal covariance of the received data in the array are proposed from which iterative algorithms are formulated to estimate spatial signatures of the sources Then greater spatial diversity is achieved by using the Spatial Smoothing in the 2L-Shape and Planar arrays In that case the estimation of the direction of arrival of the sources can not be obtained directly from the formulated algorithms The factorization of the Khatri-Rao product is then incorporated into these algorithms making it possible extracting estimates for the azimuth and elevation angles from matrices obtained using this method A distinguishing feature of the proposed tensor methods is their efficiency to treat the cases where the covariance matrix of the sources is non-diagonal and unknown which generally happens when working with sample data covariances computed from a reduced number of snapshots
Nesta dissertaÃÃo o problema de estimaÃÃo de assinaturas espaciais e consequentemente da direÃÃo de chegada dos sinais incidentes em arranjos Linear 2L-Shape e Planar à investigado MÃtodos baseados em decomposiÃÃes tensoriais sÃo propostos para tratar o problema de estimaÃÃo cega de assinaturas espaciais desconsiderando a utilizaÃÃo de sequÃncias de treinamento e o conhecimento da estrutura de covariÃncia das fontes Ao assumir que a potÃncia das fontes varia entre blocos de tempos sucessivos decomposiÃÃes para tensores de terceira e quarta ordem obtidas a partir da covariÃncia espacial e espaÃo-temporal dos dados recebidos no arranjo de sensores sÃo propostas a partir das quais algoritmos iterativos sÃo formulados para estimar a assinatura espacial das fontes em seguida uma maior diversidade espacial à alcanÃada utilizando a tÃcnica Spatial Smoothing na recepÃÃo de sinais nos arranjos 2L-Shape e Planar Nesse caso as estimaÃÃes da direÃÃo de chegada das fontes nÃo podem ser obtidas diretamente a partir dos algoritmos formulados de forma que a fatoraÃÃo do produto de Khatri-Rao à incorporada a estes algoritmos tornando possÃvel a obtenÃÃo de estimaÃÃes para os Ãngulos de azimute e elevaÃÃo a partir das matrizes obtidas utilizando este mÃtodo Uma caracterÃstica marcante dos mÃtodos tensoriais propostos està presente na eficiÃncia obtida no tratamento de casos em que a matriz de covariÃncia das fontes à nÃo-diagonal e desconhecida o que geralmente ocorre quando se trabalha com covariÃncias de amostras reais calculadas a partir de um nÃmero reduzido de snapshots
9

Hibraj, Feliks <1995&gt. "Efficient tensor kernel methods for sparse regression." Master's Degree Thesis, Università Ca' Foscari Venezia, 2020. http://hdl.handle.net/10579/16921.

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Recently, classical kernel methods have been extended by the introduction of suitable tensor kernels so to promote sparsity in the solution of the underlying regression problem. Indeed, they solve an lp-norm regularization problem, with p=m/(m-1) and m even integer, which happens to be close to a lasso problem. However, a major drawback of the method is that storing tensors requires a considerable amount of memory, ultimately limiting its applicability. In this work we address this problem by proposing two advances. First, we directly reduce the memory requirement, by introducing a new and more efficient layout for storing the data. Second, we use a Nyström-type subsampling approach, which allows for a training phase with a smaller number of data points, so to reduce the computational cost. Experiments, both on synthetic and real datasets, show the effectiveness of the proposed improvements. Finally, we take care of implementing the code in C++ so to further speed-up the computation.
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Rabusseau, Guillaume. "A tensor perspective on weighted automata, low-rank regression and algebraic mixtures." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4062.

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Ce manuscrit regroupe différents travaux explorant les interactions entre les tenseurs et l'apprentissage automatique. Le premier chapitre est consacré à l'extension des modèles de séries reconnaissables de chaînes et d'arbres aux graphes. Nous y montrons que les modèles d'automates pondérés de chaînes et d'arbres peuvent être interprétés d'une manière simple et unifiée à l'aide de réseaux de tenseurs, et que cette interprétation s'étend naturellement aux graphes ; nous étudions certaines propriétés de ce modèle et présentons des résultats préliminaires sur leur apprentissage. Le second chapitre porte sur la minimisation approximée d'automates pondérés d'arbres et propose une approche théoriquement fondée à la problématique suivante : étant donné un automate pondéré d'arbres à n états, comment trouver un automate à m
This thesis tackles several problems exploring connections between tensors and machine learning. In the first chapter, we propose an extension of the classical notion of recognizable function on strings and trees to graphs. We first show that the computations of weighted automata on strings and trees can be interpreted in a natural and unifying way using tensor networks, which naturally leads us to define a computational model on graphs: graph weighted models; we then study fundamental properties of this model and present preliminary learning results. The second chapter tackles a model reduction problem for weighted tree automata. We propose a principled approach to the following problem: given a weighted tree automaton with n states, how can we find an automaton with m
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Books on the topic "Tensors methods":

1

Farrashkhalvat, M. Tensor methods for engineers. New York: Ellis Horwood, 1990.

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McCullagh, P. Tensor methods in statistics. London: Chapman and Hall, 1987.

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Borg, Sidney F. Matrix-tensor methods in continuum mechanics. 2nd ed. Singapore: World Scientific, 1990.

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Jeevanjee, Nadir. An introduction to tensors and group theory for physicists. New York: Birkhäuser, 2011.

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Montangero, Simone. Introduction to Tensor Network Methods. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-01409-4.

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Griffith, J. S. The irreducible tensor method for molecular symmetry groups. Mineola, NY: Dover Publications, 2006.

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H, Pulliam Thomas, and Research Institute for Advanced Computer Science (U.S.), eds. Tensor-GMRES method for large sparse systems of nonlinear equations. [Moffett Field, Calif.]: Research Institute for Advanced Computer Science, NASA Ames Research Center, 1994.

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Greenblatt, Seth A. Tensor methods for full-information maximum likelihood estimation: Unconstrained estimation. Reading: University of Reading. Department of Economics, 1992.

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1968-, VanPool Todd L., and VanPool Christine S. 1969-, eds. Essential tensions in archaeological method and theory. Salt Lake City: University of Utah Press, 2003.

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Giraldo, Francis X. An Introduction to Element-Based Galerkin Methods on Tensor-Product Bases. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-55069-1.

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Book chapters on the topic "Tensors methods":

1

Chaves, Eduardo W. V. "Tensors." In Lecture Notes on Numerical Methods in Engineering and Sciences, 9–144. Dordrecht: Springer Netherlands, 2013. http://dx.doi.org/10.1007/978-94-007-5986-2_2.

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Dubrovin, B. A., S. P. Novikov, and A. T. Fomenko. "Tensors: The Algebraic Theory." In Modern Geometry — Methods and Applications, 145–237. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-4398-4_3.

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Lohmann, Christoph. "Limiting for tensors." In Physics-Compatible Finite Element Methods for Scalar and Tensorial Advection Problems, 151–210. Wiesbaden: Springer Fachmedien Wiesbaden, 2019. http://dx.doi.org/10.1007/978-3-658-27737-6_5.

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Dubrovin, B. A., S. P. Novikov, and A. T. Fomenko. "The Differential Calculus of Tensors." In Modern Geometry — Methods and Applications, 238–316. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-4398-4_4.

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Chaves, Eduardo W. V. "The Objectivity of Tensors." In Lecture Notes on Numerical Methods in Engineering and Sciences, 269–84. Dordrecht: Springer Netherlands, 2013. http://dx.doi.org/10.1007/978-94-007-5986-2_5.

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Wiśniewski, K. "Operations on tensors and their representations." In Lecture Notes on Numerical Methods in Engineering and Sciences, 6–20. Dordrecht: Springer Netherlands, 2009. http://dx.doi.org/10.1007/978-90-481-8761-4_2.

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Mardal, Kent-André, Marie E. Rognes, Travis B. Thompson, and Lars Magnus Valnes. "Introducing Directionality with Diffusion Tensors." In Mathematical Modeling of the Human Brain, 81–96. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-95136-8_5.

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Winholtz, R. A., and A. D. Krawitz. "Methods for Depth Profiling Complete Stress Tensors Using Neutron Diffraction." In Advances in X-Ray Analysis, 253–64. Boston, MA: Springer US, 1994. http://dx.doi.org/10.1007/978-1-4615-2528-8_33.

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Zhang, Yedi, Fu Song, and Jun Sun. "QEBVerif: Quantization Error Bound Verification of Neural Networks." In Computer Aided Verification, 413–37. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-37703-7_20.

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AbstractTo alleviate the practical constraints for deploying deep neural networks (DNNs) on edge devices, quantization is widely regarded as one promising technique. It reduces the resource requirements for computational power and storage space by quantizing the weights and/or activation tensors of a DNN into lower bit-width fixed-point numbers, resulting in quantized neural networks (QNNs). While it has been empirically shown to introduce minor accuracy loss, critical verified properties of a DNN might become invalid once quantized. Existing verification methods focus on either individual neural networks (DNNs or QNNs) or quantization error bound for partial quantization. In this work, we propose a quantization error bound verification method, named , where both weights and activation tensors are quantized. consists of two parts, i.e., a differential reachability analysis (DRA) and a mixed-integer linear programming (MILP) based verification method. DRA performs difference analysis between the DNN and its quantized counterpart layer-by-layer to compute a tight quantization error interval efficiently. If DRA fails to prove the error bound, then we encode the verification problem into an equivalent MILP problem which can be solved by off-the-shelf solvers. Thus, is sound, complete, and reasonably efficient. We implement and conduct extensive experiments, showing its effectiveness and efficiency.
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Bahadır, Oguzhan. "Curvature Tensors of Screen Semi-invariant Half-Lightlike Submanifolds of a Semi-Riemannian Product Manifold with Quarter-Symmetric Non-metric Connection." In Mathematical Methods and Modelling in Applied Sciences, 136–46. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43002-3_13.

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Conference papers on the topic "Tensors methods":

1

Jack, David A., and Douglas E. Smith. "Assessing the Use of Tensor Closure Methods With Orientation Distribution Reconstruction Functions." In ASME 2003 International Mechanical Engineering Congress and Exposition. ASMEDC, 2003. http://dx.doi.org/10.1115/imece2003-42828.

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Orientation tensors are widely used to describe fiber distri-butions in short fiber reinforced composite systems. Although these tensors capture the stochastic nature of concentrated fiber suspensions in a compact form, the evolution equation for each lower order tensor is a function of the next higher order tensor. Flow calculations typically employ a closure that approximates the fourth-order orientation tensor as a function of the second order orientation tensor. Recent work has been done with eigen-value based and invariant based closure approximations of the fourth-order tensor. The effect of using lower order tensors tensors in process simulations by reconstructing the distribution function from successively higher order orientation tensors in a Fourier series representation is considered. This analysis uses the property that orientation tensors are related to the series expansion coefficients of the distribution function. Errors for several closures are investigated and compared with errors developed when using a reconstruction from the exact 2nd, 4th, and 6th order orientation tensors over a range of interaction coefficients from 10−4 to 10−1 for several flow fields.
2

Moore, Joan G., Scott A. Schorn, and John Moore. "Methods of Classical Mechanics Applied to Turbulence Stresses in a Tip Leakage Vortex." In ASME 1995 International Gas Turbine and Aeroengine Congress and Exposition. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/95-gt-220.

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Moore et al. measured the six Reynolds stresses in a tip leakage vortex in a linear turbine cascade. Stress tensor analysis, as used in classical mechanics, has been applied to the measured turbulence stress tensors. Principal directions and principal normal stresses are found. A solid surface model, or 3-d glyph, for the Reynolds stress tensor is proposed and used to view the stresses throughout the tip leakage vortex. Modelled Reynolds stresses using the Boussinesq approximation are obtained from the measured mean velocity strain rate tensor. The comparison of the principal directions and the 3-d graphical representations of the strain and Reynolds stress tensors aids in the understanding of the turbulence and what is required to model it.
3

Kannan, Ravindran. "Spectral methods for matrices and tensors." In the 42nd ACM symposium. New York, New York, USA: ACM Press, 2010. http://dx.doi.org/10.1145/1806689.1806691.

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Najafi, Mehrnaz, Lifang He, and Philip S. Yu. "Outlier-Robust Multi-Aspect Streaming Tensor Completion and Factorization." In Twenty-Eighth International Joint Conference on Artificial Intelligence {IJCAI-19}. California: International Joint Conferences on Artificial Intelligence Organization, 2019. http://dx.doi.org/10.24963/ijcai.2019/442.

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With the increasing popularity of streaming tensor data such as videos and audios, tensor factorization and completion have attracted much attention recently in this area. Existing work usually assume that streaming tensors only grow in one mode. However, in many real-world scenarios, tensors may grow in multiple modes (or dimensions), i.e., multi-aspect streaming tensors. Standard streaming methods cannot directly handle this type of data elegantly. Moreover, due to inevitable system errors, data may be contaminated by outliers, which cause significant deviations from real data values and make such research particularly challenging. In this paper, we propose a novel method for Outlier-Robust Multi-Aspect Streaming Tensor Completion and Factorization (OR-MSTC), which is a technique capable of dealing with missing values and outliers in multi-aspect streaming tensor data. The key idea is to decompose the tensor structure into an underlying low-rank clean tensor and a structured-sparse error (outlier) tensor, along with a weighting tensor to mask missing data. We also develop an efficient algorithm to solve the non-convex and non-smooth optimization problem of OR-MSTC. Experimental results on various real-world datasets show the superiority of the proposed method over the baselines and its robustness against outliers.
5

"Session MA1b: Tensors methods in signal processing." In 2010 44th Asilomar Conference on Signals, Systems and Computers. IEEE, 2010. http://dx.doi.org/10.1109/acssc.2010.5757212.

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Zhang, Yuan, and Regina Barzilay. "Hierarchical Low-Rank Tensors for Multilingual Transfer Parsing." In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing. Stroudsburg, PA, USA: Association for Computational Linguistics, 2015. http://dx.doi.org/10.18653/v1/d15-1213.

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Babudzhan, Ruslan, and Oleksii Vodka. "Comparison of Glyph Visualization Methods for Structural Stress Tensors." In 2021 IEEE 2nd KhPI Week on Advanced Technology (KhPIWeek). IEEE, 2021. http://dx.doi.org/10.1109/khpiweek53812.2021.9569986.

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Polajnar, Tamara, Luana Fagarasan, and Stephen Clark. "Reducing Dimensions of Tensors in Type-Driven Distributional Semantics." In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP). Stroudsburg, PA, USA: Association for Computational Linguistics, 2014. http://dx.doi.org/10.3115/v1/d14-1111.

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Yang, Chaoqi, Cheng Qian, and Jimeng Sun. "GOCPT: Generalized Online Canonical Polyadic Tensor Factorization and Completion." In Thirty-First International Joint Conference on Artificial Intelligence {IJCAI-22}. California: International Joint Conferences on Artificial Intelligence Organization, 2022. http://dx.doi.org/10.24963/ijcai.2022/326.

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Low-rank tensor factorization or completion is well-studied and applied in various online settings, such as online tensor factorization (where the temporal mode grows) and online tensor completion (where incomplete slices arrive gradually). However, in many real-world settings, tensors may have more complex evolving patterns: (i) one or more modes can grow; (ii) missing entries may be filled; (iii) existing tensor elements can change. Existing methods cannot support such complex scenarios. To fill the gap, this paper proposes a Generalized Online Canonical Polyadic (CP) Tensor factorization and completion framework (named GOCPT) for this general setting, where we maintain the CP structure of such dynamic tensors during the evolution. We show that existing online tensor factorization and completion setups can be unified under the GOCPT framework. Furthermore, we propose a variant, named GOCPTE, to deal with cases where historical tensor elements are unavailable (e.g., privacy protection), which achieves similar fitness as GOCPT but with much less computational cost. Experimental results demonstrate that our GOCPT can improve fitness by up to 2.8% on the JHU Covid data and 9.2% on a proprietary patient claim dataset over baselines. Our variant GOCPTE shows up to 1.2% and 5.5% fitness improvement on two datasets with about 20% speedup compared to the best model.
10

Ko, Ching Yun, Rui Lin, Shu Li, and Ngai Wong. "MiSC: Mixed Strategies Crowdsourcing." In Twenty-Eighth International Joint Conference on Artificial Intelligence {IJCAI-19}. California: International Joint Conferences on Artificial Intelligence Organization, 2019. http://dx.doi.org/10.24963/ijcai.2019/193.

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Popular crowdsourcing techniques mostly focus on evaluating workers' labeling quality before adjusting their weights during label aggregation. Recently, another cohort of models regard crowdsourced annotations as incomplete tensors and recover unfilled labels by tensor completion. However, mixed strategies of the two methodologies have never been comprehensively investigated, leaving them as rather independent approaches. In this work, we propose MiSC ( Mixed Strategies Crowdsourcing), a versatile framework integrating arbitrary conventional crowdsourcing and tensor completion techniques. In particular, we propose a novel iterative Tucker label aggregation algorithm that outperforms state-of-the-art methods in extensive experiments.

Reports on the topic "Tensors methods":

1

Bouaricha, A. Tensor methods for large, sparse unconstrained optimization. Office of Scientific and Technical Information (OSTI), November 1996. http://dx.doi.org/10.2172/409872.

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Schnabel, Robert B., and Ta-Tung Chow. Tensor Methods for Unconstrained Optimization Using Second Derivatives. Fort Belvoir, VA: Defense Technical Information Center, July 1989. http://dx.doi.org/10.21236/ada213642.

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Schnabel, Robert B., and Paul D. Frank. Solving Systems of Nonlinear Equations by Tensor Methods. Fort Belvoir, VA: Defense Technical Information Center, June 1986. http://dx.doi.org/10.21236/ada169927.

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Mayo, Jackson R., and Tamara Gibson Kolda. Shifted power method for computing tensor eigenpairs. Office of Scientific and Technical Information (OSTI), October 2010. http://dx.doi.org/10.2172/1005408.

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Bouaricha, A., and R. B. Schnabel. Tensor methods for large sparse systems of nonlinear equations. Office of Scientific and Technical Information (OSTI), December 1996. http://dx.doi.org/10.2172/434848.

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Haass, Michael Joseph, Mark Hilary Van Benthem, and Edward M. Ochoa. Tensor analysis methods for activity characterization in spatiotemporal data. Office of Scientific and Technical Information (OSTI), March 2014. http://dx.doi.org/10.2172/1200656.

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7

Chow, Ta-Tung, Elizabeth Eskow, and Robert B. Schnabel. A Software Package for Unconstrained Optimization Using Tensor Methods. Fort Belvoir, VA: Defense Technical Information Center, December 1990. http://dx.doi.org/10.21236/ada233989.

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Bader, Brett William. Tensor-Krylov methods for solving large-scale systems of nonlinear equations. Office of Scientific and Technical Information (OSTI), August 2004. http://dx.doi.org/10.2172/919158.

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Schnabel, Robert B., and Brett William Bader. On the performance of tensor methods for solving ill-conditioned problems. Office of Scientific and Technical Information (OSTI), September 2004. http://dx.doi.org/10.2172/919164.

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Manke, J. A tensor product B-spline method for numerical grid generation. Office of Scientific and Technical Information (OSTI), October 1989. http://dx.doi.org/10.2172/5005256.

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